Calculating machine



My 10, 1949. J'. R. BowMAN ET AL 2,469,628

CALCULATINQ MACHINE A Filed April 27, 1945 v 5 Sheets-Sheet 1 l' l l HU Q www

May 10, 1949.. JQ R. owMAN l-:T AL 2,469,628

CALGULATING MACHINE v Filed Aprii 27, 1945 Y 5 'sheets-sheet 2 Il v l A c: INPUT voLTAega 15OV. +`15OV.

RESPONSE or* RcTl/FIBR O GURRENT SUPPLY UNIT 115V., 60A k FROM AMPLIFmRl FROM oscl'LLpAToR .JOHN RBOWMAN RALPH 'Tl STJINBGK May l0, 1949. J. R. BOWMAN ET AL l 2,469,628

CALCULATING MACHINE I Filed April 27, 1945q 5 Sheets-Sheet 3 B POWER SUPPLY 36* fac. 15C. INP UT v FRGM OSCILLATOR O O TP U T l l D.C. INPUT VOLT Lisa. Ef 52 v BMP F11-f.v

JOHN R. BQWMAN RALPH T. STEIN BACK M35' 10, 1949- J. R. BOWMAN ET A1. 2,469,628

CALCULATING MACHINE Filed April 27, ,1945 l 5 Sheets-Sheet 4 I l I I I I I l I I Y Ik I l Arf' I I JOHN R.. BOWMAN RALPH T-.STEINBAGK- Patented May 10, 1949 burghjl Pai); assignors; to

Gulf Research & De-

velopment Company;Bittsbm'ghiazcacorpora@ tion' of Delawaren3 This invention relates tocalculatingmachines and -wmoref particularly'to :electronic calculatingsz; machines adapted=for use in the solution of rlinearr. r simultaneousl algebraic equations. f

In the priorlandf' :copencling application Serial Nof479g790 lel--March- 19`, 1943,V by:` Johnl :Rg Bowmanfon Calculating: machines, thereffis olisf4 closed and' claimed an 'aigipai-atu'sfl for' an eleca A tronic icalculating" machine for `use ein'r ythe soluftiene-f linear; simultaneous algebraicequaticnsz: 1011:

The-present'*application@concerns fanimprovecla;y

onerwhichfis :simpler and hasgreater-accuracy:fl`

Slutionf'fof- 'systeinsaof i 'linear' algebraic f equa'- l computation; f Elementary' methods'v ofA felirrlin'aeA tion or substitution arelquiteisatisiactory for sys'1 temsfhavingi not more "than 'fourvariables: For larger systems, fhowever; the calculations become extremely laborious,v 4as 'the' number of arithmeti' calf opera-tions" required increases approximately ,-1 as n.11! for n variables; exact solution offa sys temg-of20variables requires more than .10201 'operi ations. Manyiye-ars ago Gauss poi-ntedlout that=- anypr-cbl'emv incomputationlcan, theoretically,v =be 25 reduceclito :solut-ioncof/i a `linear asystemf ofVV equav I. tions which fact -hasi'subs'equently from' a practi'f cal'standpoint been 'accepted' as true For lar-gesystems; iexcept those with many termsmis-s'ingw the-simplest"straightforward" method available is y30 thatzof Sylvester"employing` determinantsj which' f is not'readily: adaptedtoI` a conventional keyboard A' i calculating f'machine.- Practically',H systems of greater than iivefzvariables are nearly always solvedifbyfmethods of successive approximations; `35 Thse f approximate Jmethods are 'also laborious; and" frequentlyio 'not' give good. accuracy;Y they?,` are-'discussed-aisome length later. The-numeri# caln solution'of systems oequations Xthus presents441V a problem'requiringa specialztype of calculating-"40 machineri- Fui'tlrer',`:since in' `thepast; solutinnwoi-` large-j systems";of"simultaneousf'equationsfwasdiicult or impossible; little .attention 'was paid to* the.' redubtionzof problems to such' systems? doubt-l 45 lessl the4 availability of" a machine" of the present 1 type will" "stimulateinte'rest" in" new transformaf tions; that'. willlextend the .use of 'the 'instrument beyond; applicationsnow obvious.

Several; 'difficult' mathematical operations -are:;150" equivalentlto' solutionzoia linear algebraic system: Oneof'these isthe expansion olf/functions'. L in series; withlan n variable device, ln. termstarer readilyljdetermined.1Harmonic" analysisliis'ga specialecase of'thi's; and wouldiirovde an excel' 55:'

2 lent justification forthe machine ralone. Numara ous.;deviceshavef-laeenabuilt:speciflcallyrfor:hartemomo-analysis, .somekof'themiveryxelaborate'and'gf none 1# as Ypotentially if', accuraterfas @this Lfldevice: is. Otheriusetu'l: expansions'arefseries 1of::powers';fex c potentialsjserrox -;functi'ons, yspherical harmonicsa andBesseltffunctionszf r. These :operations :are thenbest rway offanaly'zingfandiageneralizing empirical if functional-forma aand-theyfoccur Nery frequentlyfv' in pure"and=applied=-physics.

Th' present device-:isialsoi-capable-of integral;'--'-4 ing= ordinary# linear@ differential; equations; r= Here; i thefgivenfequatioltlisregarded yas3 a -nite diffe enceequati-on'with the1incrementsveiysmallzjf` `pointsfof .theiv 'integra-lf areaobtainedl on-:ra fsingle--v pas'srlthroughfwtlie instrument-,i Fand-J 'any 1-nurr1bez e of passesfmayb made'fusinfgf.the-last=pointfofetliefv lasi'f-setasV tlefbundary-"point' foifthe --next--^ se Good accuracywisi'obtainabie, since:'-theintervals11.-y

20"-betweenf-the =pointsffcan=be made/livery:smallL 'with out incur-ringlaborious` calculation:

Linear iintegralfequationsf can likewise Lbe soly`ed --'2 numericallin` Repeatedaiise ofA the machine will l i againfincrease'atlreaccuracy of the" solutionfbutf not'-assatisfactorilyashini -caser of differential'- equations" becauseL the Jr1. 'pointsselected for i the independent variable fmustI necessarily coverwtheventire range-mf` integration ra-ther 'unif( r1i'1ly-y HoweveIg-Yif *n si about #151 foreigreater-sufficient 'accuracy' will:loef-obtainedfor=lnearlyl all` practical f worksl Th applicationj of'the--present'machine Y in this connectioneis essentially -Va finite' case off Y* then welllknown 'Fx-e'dhlnr processifa more ren'ed approximation; dependin'gfon* the reduction 'of theequation-to a linear *systemi'has-been' described '1 recentlyg iir1'-"C1"cn-xt, 'J. Math Physaf Mass. `-I1:s,t:. Tech-.1;A 195134"t (19"l0I)"f."-`l Thseappllcationsare-0F particular"interestatpresent because -of`- the ace tive modern interest in integrall equations` in man branches:offscienceand engineering.1v

Correlatlonwof HataWbyJthe method 'of' least squareswalways involves the;l `solutiorroflinearf` algebfaicisystems: This-isla particularly'limporirr- 'tant'application',h and4 will appear:k in nearlyevery 1 typenf Lexperimental"numerica1''work from sur-A veying topsychologyr" 4 Numeri-cal solutions' of'l linear Systems' arefoff' importance" inra few highly theoretical lines,` of endeavor?goodfexamples: are ling grzoup theory in' 'pure'math'ematics *and in'ithe calculation. ofiwave: 'l functions' bytlie' Hartreeemethod jof -=Quantu`m la, mechanics; i

n."leictrical4 networks furnish 3a vgreat, y.class` of practic'alf'probl'ems of such importance thai-seyneral specialized machines, more or less` equivalent to the present device, have been built. The mathematical statement of these problems is directly in the form of linear algebraic equations, arising from the application of Kirchhoifs and Ohms laws, and requires no reduction for solution by the present device.

Stress-strain problems in connection with structural units comprise an important division of mechanical and civil engineering. These lead, in the more complicated cases, to the determination of several unknowns in a linear system. Machines, dams, bridges, airplanes, and buildings are just a few examples of design problems where this device might help.

Step-Wise countercurrent processes, as widely used in chemical engineering, are easily treated rigorously by the present type of machine. These include rectification, as in a bubble plate column, absorption, adsorption, extraction and chemical reaction processes.

Chemistry oliers at least two general types of problems in which the device would be useful, equilibrium and kinetic calculations. The watergas reaction is a good example of the rst. Here, ve components are mutually interconvertible through four reactions; if the equilibrium constants are known, as they are in this case, the equilibrium compositions may be calculated at any temperature by solution of a linear algebraic system. The inverse problem is also readily treated; if the compositions are known, the equilibrium constants can generally be calculated, even for rather complex systems. Kinetic calculations are formally similar. In these problems, one has to deal with reaction rates, and if the reaction rate constants are known, the progress of the reactions in a complex system can be followed with this device. Gas-phase combustion is an example of this type of problem.

The immediate response of the present device, and the ease with which the coeicients and constant terms may be adjusted permits its use for control work. Consider, for example, the use of the mass spectrograph as a gas analyzer for control of a still. Each component gives a spectrum of several peaks. The patterns for the different components are all different, but the peaks are more or less superimposed. By selection of n suitable peaks, the composition of the gas with respect to 'n components may be obtained continuously, and used for indication, control, or recording. More generally, any mixture may be analyzed by measurement of n independent physical properties, and if these properties are linear functions of the compositions, the device will give the analysis continuously. Density, refractive index, vapor pressure, optical activity, and absorption spectrum are a few of the properties that might be used.

Only a few machines for solving systems of linear equations have been built in the past. The earliest successful one is a six-variable instrument described by Mallock; R. R. M. Mallock, Proc. Roy. Soc., A140, 457 (1933) which depends on the A.C. flux in numerous small transformers having multiple coils. The principle of that machine is very simple, but the accuracy is poor, and becomes increasingly so as the number of variables is increased. Rather complicated compensating circuits are required to correct for the losses in the transformers. Mention is made in the original paper that the Cambridge Instrument Co. has built a machine of this type for ten variables, but no deiinite performance data are given for it.

The only other instrument designed strictly for the present application that has been fully described is that built at the Massachusetts Institute of Technology and described in detail by Wilbur, Jour. Franklin Inst., 222 715 (1936). This is a mechanical type for nine variables, depending for action on simple geometrical properties of levers. The accuracy reported is good, but the machine is diicult to set for a problem and considerable time is required for indication of the solution. rlihe cost of this type is great; many small pulleys, levers, ball-bearings, steel bands and micrometer screws are required, and the machine takes up a considerable amount of space.

A network analyzer has been built by Westinghouse and described by Travers and Parker in The Electric Journal, page 3, (May 1930). This is a machine specifically designed for solving problems in electrical networks by a process which practically amounts to setting up of scale models of the circuits and measuring their performance. It is not strictly equivalent to the proposed type of machine, inasmuch as each equation must have only a limited number of terms, though the number of Variables may be large. Furthermore, it is not suitable for solving problems given in the usual algebraic form, belcause the discovery of the network corresponding to the given problem is not a straightforward operation. rIhe Westinghouse machine is a very large one; its control boards fill three walls of a room about 20 x 20 feet, and have about 8G00 switches, several plug boards with hundreds of jacks, and numerous other controls. The makers built the machine for their own research use, not for commercial production.

None ci the machines described could feasibly be adapted to solution of systems as large as the proposed one will solve, and the proposed one is superior to all with respect to speed, accuracy, cost and size.

Therefore, the primary object of this invention resides in the provision of a exible electronic calculating machine that is adapted for solving systems of linear algebraic equations which include those of the types mentioned above.

More speciiically, it is the object of the present invention to provide an electronic machine for determining the roots of n variables related by n linear algebraic equations, where n is any integer greater than unity, comprising n ampliers giving an alternating current response to a direct current input, n2 rectiers supplying direct current, n2 voltage dividers adjustable to be proportional to the coeiiicients of the system of equations, n sources of direct current voltage adjustable to be proportional to the constant terms of the system of equations, these circuit units being connected in n circuits, each traceable from ground through a voltage supply, through 11, of the parts of the voltage dividers in series and into the ampliiier, the input of which has a grounded return, each of -the voltage dividers being supplied with current from one and only one of the rectiers, and the output of the mth amplier, where m runs from one to n, so connected as to supply to the mth rectiiers of all the said circuits, and at least one of the said circuits being provided with current measuring devices in series with rectifiers giving indications that can be read as desired roots of the system of equations.

Another object of this invention is the improvement of the device described in co-pending application; Serial '.No..=. 479,9 T0; which comprisesA the substitution of voltage dividers for adjustable-resistors.v Thexusexof the voltage dividersr'removes the need xfi'nreguiation of. the currentlsupplied to them .so :that the internal resistance ofthe' cur'- rent supplies neednot-be large. Thispermits the employment .of simplerectiii'er circurtsas current suppl-ies in'place of fthe more complex regulated currentsupplyunits required for the device described the above-mentioned application.

Another` object 4-oithis 1 invention resides inthe provision ofelectr-onic calculating machines which employ the principlespi fullydegenerative feedback.

Still another object of this invention resides in` the provision of an electronic calculating machine adapted for use in solving a system of n simul# taneousalgebraic equations havin-g11n-variables which will-indicate directly andc-ontinuously the roots thereof. y

This invention iur-ther contemplates:the pro-` vision of an electronic calculating machine for the solution orl systems of algebraic equations that issmall, compact, economical to manufacture, and easyfto operate.

Other robjects and advantages of: the'presentv invention Willbeccrne apparent from the follow#- ing detailed descriptionWhenconsidered vS/"it-h the draw-ingsin which:

Figure 1 isa block' diagramv oi a4 three-variable machine Which-'serves to illustratetheprinc'rple ot'he iirventioniwhich isdirectedto an 91, variable mach-ine;r

'Figure .251s a diagrammatic illustration oflanl amplier illustratingtheqprinciples oi fully degenerative. fee'dback `in. the simplest form 'used in this machine;

Figure 3 is ade'tailed Wiring diagram of' one of' the 'n2 rectiiier units ofthe type usedlinthe novelv machine 'forming the subject matter of th'isapplication;

'Figure 4 is a curve which has been-plottedwith direct current output' as ordinates and A.'C'. volt'.- age input as. abscissae representingr the response of each of the rectifier current supply units.;.

Figure 5 is a detailed Wiringdiagram illustrating the circuit for one of the n constant term. voltage supplies;

Figure 6' is a A 'letailed circuit diagram of one-cf the n amplifier units",

Figure "7 isavcurve'that has "been rplotted 'with' root-mean sqirareA-.-C. -output voltage asfordi# nates and D.C. input vVoltage 4'as 'abscissaewhich illustrates theY response to one ofthe amplier units;

Figure 8 isa Wiring diagram representing'the; powersupply orthe ampliers; and

Figure 9v isv a .block-.diagram oi'a two-variable machine constructedV according. tothe presentinvention and showinggin detail thev connections to the A.C; oscillator. y

Figure 1.0 is a wiring diagram ofv a two Variable machine of Figure 9 andshows in detail the wir.- ing of one equation circuit.

For. the sake of' clarity in kunderstanding ther Figuresdl, 2' and 9,*those circuits. carrying DFCL. are shown by. solid .lines thuscircuitscarrying A.C. from. amplifiers to -rectiems .are shown.

f one'which l'Would handle problems not amenable By Way yof' explanation .ofr the principles in Volved in the present invention:

Let a problem be given in orreduced'to, the form 1ct-:aaai y= "0. i,-g, ='1',-2", ,11,v

The-(proposed machine recognizes the constant terms '(k's) asvoltages., the coefficients. (asi as ohmicresistances.,v and the variable (ms) ascurrents. Each equationisrepresented by a single circuit, beginning at a point on a voltage divider or, potentiometer'which may bevoltage above or below: ground, and having-.n voltage :dividersin series.. Each voltage divider is provided withian independent, oating, D.C. supply. These suppli'eswill 'establish` voltage drops across themadjustable part fof the voltage divider which are proportional to the ax terms. rThese. voltage drops, since they are in series, will add, falgebraicaflly, to' the reference voltage `corresponding 17o-the constant termto produce a voltage atthe'. end Lof'f the circuit proportional 'to the Valueof the left side ofI the equation.- Iinow, the' n2 current supplies. can be'made to supply currents such as the mthyalues are equalr to m, andv all -of' these latter4 voltages are Zero', the rootsk maybe read immediately by measurement of the currents 'and' thesyStem of equations is thereby solved'.

Again referring to Figure l, the current supplies CSv comprise rectiers Which are designedv to .give an output of direct' current', the'magnitude of'whic'h is dependent en the A.-C; supplied' them. All n2 of them are identical; Since, when the givenequations are reduced tothe canonical' form,.the appearance of 'the lvariables, kin the am termsis .such that one `variable occurs alone in one column, and since .the current outputs ofthe t current supplies correspond to the variables; the output and hence the .A.-C. supply voltages must" be equal among all the rectifiers of any one column. Full degeneration in the amplifying y system is accomplished'by causing the sum voltages of the n equations to generate the A.-C.

source voltages for .the rectifiers by columns-respectively Obviously, this arrangement Willcorrect. an. error in any diagonal am term. If the :c in.

that termis too large or small, a signal is communicated' to the. amplifier connectedto that equationrcircut, Whichgenerates a change in the A..C.` voltage. output of that amplier, .and that,

A.C voltage change. is applied directly to the direct current .supply corresponding to that a: in a.

manner to correct its value.. These current-suppliesactuate the meters F, G andH which can be calibratedv to. read directly the roots of the.

three equations. Since the current supplies are allalikeonly .onemeter is .required .for each col-v umn; This principle of .ope-ration is .disclosedinl aforementioned:application Serial No. 479,790.

Discussing the elements of themachine .separately, and vin detail .according to the embodiment r ofthe present. invention referenceismade to the. I rectifier eurent supply units :CSfof Figure 1, one.

of which is [shown in detail in Figure 3. It comprises essentially two double-rectifiers connected with their D.C. outputs in opposition and their inputs separate except for a common condenser blocked ground return. The net D.C output voltage is therefore equal to the diierence between peak voltages of the inputs. The output is provided with a plurality reversing switch 20 for selecting the sign of the coeicient when a problem is put into the machine. As shown diagrammatically in Figure 1, meters F, G, and H are placed in the current supply output leads. Such a meter is illustrated in Figure 3 at 2 I.

The type of rectifier circuit used is limited by the fact that the output must float with respect to D.C. potentials from ground. This requires that all D.C. voltages in the A.-C. supply must be blocked out either with condensers or transformers. If condensers are used, the rectifiers must be of the type that utilizes both half cycles of the A.-C. supply; thus a half wave voltage doubler can be used, but a simple half Wave rectifier could not.

In the circuit of this invention, which will be described in Imore detail in connection with Figures 9 and l0, one of the A.C. inputs to Figure 3 is supplied by a constant voltage A.C. oscillator while the other is obtained from a suitable amplifier to be described later. The direct current output, representing an :c reverses when the amplifier voltage is equal to the oscillator voltage. This direct current varies linearly with the amplier output but has a definite constant value even with no A.-C. input from the amplier. The entire D.C. circuit is free to oat. Stability is excellent and there is some self-compensation for power line voltage fluctuation since this would effect both the oscillator and the amplifier in the same way.

Two rectifier' tubes I3 and ICI are connected as indicated in Figure 3. The-se may be type 2526 or other type of double diode having independent cathodes. The heater circuits are not shown as they are of conventional form. Input condensers I I and I2 are provided to isolate the circuit with respect to D. C. The double circuits are similar, the output of tube I3 being filtered by condensers I5 and Il and resistor I9, and the output of tube I I being filtered by condensers I6 and I'I and resistor It. The output of the unit is therefore the difference between the D.C. voltages developed across resistors I8 and I9. By using high values of resistance, the current drain is kept very small so that these voltages are essentially the peak A. C. impressed on the two halves ci" the input.

An important point is that the entire circuit for the current supply unit is isolated from all D.C. potentials. Filament power is supplied through transformers and the A.C. input circuits, both input and ground return, are blocked with condensers. The current output therefore floats and may assume any D.C. potential required. In operation this may reach 900 Volts when a three-variable instrument is unbalanced, or 380 volts when it is balanced.

Response curves which have been plotted with output current vs. A.-C. supply voltage are illustrateol in Figure 4. It is apparent from Figure 1 that, if the amplifier input impedance is kept very high, the only load on the unit is merely that of the potentiometer R (Figure 1) which is constant and also quite high. Thus good stability is obtained` The response curve shown is by way of example only and may be changed to suit the particular machlnasince the particularv application of the machine controls the load involved.

The constant term voltage supply shown dagrammatically in Figure 1 is illustrated in Figure 5 and requires little description. A standard full wave doubler circuit is used which employs a type 11726 rectifier tube. The mid-point 23 of the doubler circuit is grounded at 24. The voltage dividers therefore yield outputs of -150 to +150 volts, on a linear scale, when a 60 cycle power supply of volts is used. This unit is the only one of the entire circuit that is sensitive to line voltage fluctuation. A change in line voltage will effect the indicated roots by a proportional change in the constant terms. Therefore, for a precision instrument, these voltages must be stabilized.

The amplifiers used in the present invention are of a special type which put out A. C. for a D.C. input. The manner in which these amplifiers cooperate with the current supplies to yield the proper currents is by the use of a degenerative or negative feed-back circuit. This is best explained by consideration of the simplest circuit of that type, essentially one for the solution of a single equation with a single unknown by the system of this invention. That circuit is illustrated in Figure 2. An amplifier 25 is used which has a large input resistance, a large voltage gain, and a floating output. It is of a type which gives a large A.C. output for a small D.C. input. Ignoring the condenser 21 for the present, when a D.C. voltage is applied across the points E, the amplifier 25 initially functions to give a larger A.C. voltage across its output 28. This voltage is rectied by means of the rectifier 29, and the resulting D. C. is applied across a resistor 26 in series with the input of the amplifier 25 with polarity such that it tends to buck out the input voltage initially applied across the points E. The equilibrium position of the circuit is readily calculated quantitatively. The fundamental performance of the amplifying system may be stated:

. Gain X Input Output In this case,

G(EEc) :En or E/E0=1+1/G Therefore, if the gain is large, any voltage applied at E is very nearly equal to and balanced out by the output voltage developed across the resistor 26, and the true input voltage to the amplifier remains very nearly zero.

It may be noted here that the balancing out of the applied voltage is independent of the value of the gain as long as the latter is large. Distortion and instability of the amplifier therefore have only minute effects on the performance of the circuit. Any intermediate transformations made in the amplifier network, such as in the present case the transformation to A. C. and back again to D. C. in the current supplies are of minor consequence only insofar as they may effect the gain G.

A circuit essentially similar to this one has been described by Vance, Rev. Sci. Inst., 7 489 (1936), for use as an electronic meter. For voltage measurement, a milliammeter is placed in the output circuit, and a precision resistor used to neutralize the input. The instrument is then almost wholly insensitive to line voltage fluctuation and drift in amplifier characteristics, and its accuracy is solely dependent on that of the meter and resistor, input voltages being calculated over a wide range by mere-application of Ohms law.

'For'measurement-'ofv current, the instrument is shunted with another lprecision resistor.

kof the ampli-ner used, which may be nearly infinite.

The quantitative discussion of the performance of the circuit 'givenabove concerns only static equilibrium. In operation, and in automatically adjusting itself, the dynamic performance is also important. The'mathematical treatment of the dynamic performance of the degenerative circuits has been fairly lWell developed, and is very oomplex,as described by Nyquist, Bell Syst. Tech. Jour.,l1,v 126 (1932). The results of the analysis, however, are fairly simple for the present circuit. It may have a stable 'or unstable equilibrium point; in the latter case it oscillates. Oscillation may be suppressed or eliminated entirely by the addition of the condenser 21 of Figure 2, and whenthis is done,-'the stable equilibrium state is the same as the one reached by simple calculation of static conditions.

The matter of stability of the computing machine herein described .has been investigated theoretically. The general condition for stability of feed-back circuits is derived by Nyquist, Bell Syst. Tech. Jour., .11, '126 (1932). Consider an amplier of gain G provided with a feed-back network (in this instance the rectifier current supply) .with gain.B. The quantities G and B will in general be complex, because either or both the amplifier and the feed-back network may introduce phase shift, and they may vary with the frequency. 'If allpossible values of the product GB and its conjugate (for all frequencies) are considered, it is found that the system will oscillate if the over-all in-phase gain in unity. This gives d'eiinite specifications for the amplifiersl used. The feed-'back network (viz. rectifier) is purely vresistive so that B is a real number, and therefore'the amplifiers must be so designed that their gain is less than l/B for those frequencies at which the input and output are in phase. This isreadily achieved by using high frequency yby-pass lfilters inthe input circuits of the'ampiilers, and designing the ampliers themselves for low response at high frequencies.

The factor B is also under control though it is inherently iixed by the coefcients controlling the resistors for a given vsystem of equations. Its value is under control in that it depends on the order in which the-equations are written and on the order of the variables in the equations. A convenient suiicient condition for stability (assuming that the amplifier meets the abovementioned requirement) is that B be negative. This condition is` also very nearly necessary because it must'be always less than l/G when both B and G are real, and G is usually a large number. This reduces to the simple rule that a system of equations can give a stable solution in the machine if the matrix of the coeiiicients is such that all diagonal minors have the same sign. This rule concerns the stability of the machine once the solution has been attained. Nearly all systems of equations lmet vin practice fulfill this condition or may be transformed to ones that do by writing them so that the largest coefficients lie on the diagonal ofthe matrix.

A more complete study involves also the dynamic manner in` which the machine approaches the solution yequilibrium state. 'I'his involves a solution of ther general differential' yequationsfor the dynamic actionof amachine for variables. The system of differentialf equations is' linear and 'maybe solved. `The^resulting vcriterion isthat the machineA will solveariy system ofV equations for which the real parts of the roots of the 'characteristic lequationfof y'tl'iematr'ix' of' co'eilicients 'are positive. "Equations not-:satisfying this condition canv alwaysbe" transformed into vcries that do satisfyit. "See paper viyW."VJParker on "Ihje limits to the characteristic roots 4v'of 'a matrix, Duke` Math. 3., '10, '-i79"482 '(1943). The transformation oan usually be determined by inspection. lUpon introducing'suchlasystem of equations into themachine solution. n

v The ampliiiers fused 1in: the* machine `:are f of a rather specialitype. *"Ti'iey 'must respond toja Df-C. inputvof lowyoltageand supply an "A.C. output. In this casethevhigh gain implies that for a small D.`C.'lnput'tlie'ymust supply a very large A.'C. output. -Linearity 'andv distortion are wholly unimportant. Furthermore, ytheyl must show zero output'when the'inp-utsgnal is negative, and .a'largeigain as the signal goes slightly positive. v`:Finally the drift of the zero point must be made verysmall. The vre'ctiiers maybe re'- garded as parallel output stages of the' amplifiers.

`All'this has been accomplished inthe present invention bythe-circuit given in Figure 6, which is ay detailed circuit of "an 'amplifier' unit "marked A in Figures "i andf9; Startin'g'withltheswitch 3i (also shown vin Figure "1), kth'e`D.-"C'. input is to oneY ofthe v'signalfgrids 35 of'a'mixer tube 32, such -as a type y'61.7. 'An input iter is provided by resistor 33 V,and condenser 34. iliriotiier signal grid ESfis'suppIied with"A.^C.-from an oscillator throughk avcoupling device consisting of condenser 36 and grid resistor 31. yTheother'connections to tube 32 are in' a oonventionalresistance'coupled amplifier, except `that f the 'load' -resistance .'38 verylargef250',000 ohms) and' thev voltage onthe screen grids very small (about +5 Volts) Tub-e 32 'deveiopsacross its plate resistorss' an AmC. potential Iwhich 'is proportional to the D.C. input within certain'iirnits. This A. C.i's' further iampi-ined by vmeans of "a power 'tute 39'. which'may'be a type-idilio. 'Converitionfai*resist-` ancevv coupling isprovfded by'cor'ideris'er 4`0` and re sistor 4|. Other connections to`tube39 are conventional. vOutput is 'obtained through transformer 42.

In operation the outputof tube 32` consistsy of A. C. regulated by the D50. inputvoltage. The tube cuts'off'sharply'when the input-LD. C.4 goes isass'urdof a stable only slightly. negative y(about 0`5- volt). The

zero Vpoint V`is characteristicof the -tube and is stable within a small range. A vtypical lresponse curve for `one of the amplifiers visfshown in Figure', this being'by wayofl example-,only since-itI may vary depending on'the sizeof thel'machine and' the number yof` currenty supplies to be conr` nected to the ampli'iier output.

The amplifiers` are operated from acomm'on power supply shown Iin detail in Figure 8. Since this unit has no unusual features except that the iiltering as provided by the network shown com-- prising the iron core*in'du-ctancesY 4'9 'andll 'and the condensers A5I and-"52" is preferred for such circuits. This better'thanusnall filtering isinecessary because anyx ripple remaining will appear as A.C. output voltage lfrom theampllers even when the input'is negative., and' this will reduce the effective range offth'e "current supplies; In

' this ligure 'the 'source of supply isshown 'as`1`10 ll volts A. C. and is impressed across the primary winding of a transformer 53 .whose secondary is divided into a plurality of windings for supplying the amplifier filaments, the heater voltage for the rectifier tube 54, and the power supply to the rectier tube.

In Figures 9 and 10 we have shown for a two variable machine constructed according to the present invention detailed connections including those by which A. C. is supplied to the various units by the oscillator. The oscillator OSC mai be any conventional oscillator having sufficient power to adequately supply all of the current supplies CS and all of the ampliers A. Its frequency is not important; we have found that a frequency of 1000 C. P. S. is advantageous. Other elements of Figure 9 have the same designation as previously explained. Figure 10 shows in detail the wiring for one equation circuit, i. e. the top row. The other equation circuits or rows are similarly connected. Filament heater and plate supply power are supplied to all units in a conventional manner and for simplicity these circuits are not shown. The circuit of Figures 9 and 10 may be extended toa machine of any number of variables in an obvious manner, the two variable machines being shown only for simplicity of illustration.

In operation, to set the machine for a problem involving a system of simultaneous algebraic equations of n variables, an instrument having at least n elements is used. The n2 coeiiicients are set by means of the variable voltage dividers R illustrated in Figures 1 and 9 and the ln constant terms are set by means of the dividers S on directly calibrated dials. The roots of the equations are then read on the small meters such as those shown at F, G and H, in the threeelement machine illustrated in Figure 1, or F and G on the two element machine of Figure 9, continuously and essentially instantaneously.

It is frequently convenient to make transformations on given systems of equations before setting themin the machine for solving. For example; if one column of coeiicients are all relatively small, and the variable associated with that column is known to be large, these coefficients may be increased by the same factor. A similar treatment may be used when one column of coeicients is relatively large.

The results of action of the machine need not necessarily be indicated on the meters as the output can be used for recording and/or control, as well as for indication.

The instant invention is sumciently flexible that it may also be used for practicing J acobis method for the solution of systems of algebraic equations. Jacobi proposed a method similar to that of Gauss and Seidel and mathematically equivalent to it. In the use of his method, corrected values of the assumed roots are sought, rather than the explicit corrections. This method, however, has received little attention from calculators.

One may write the given system of equations in the form Then assumed values are substituted for the s in the left side; a knowledge of the approximate values of the roots is of considerable help. The value of mi is now adjusted so that the first equation is satisfied; this is readily done, as it amounts merely to solving the first equation regarded as a single linear equation in one unknown. The

value of :ri so found is now substituted in all the1v equations, replacing the assumed value. Similarly now, the second equation is solved for 3:2. and the value so found substituted in all the equations. When this value is substituted in the rst equation, that one will, in general, no longer be satisfied, but the second one will. The process is then continued through the n variables, solving the ith equation for :ci in sequence, and substituting in the entire system the values found. When the process has been completed through the entire set of variables, the set of values in effect can be regarded as the rst approximation to the roots comprising the solution. The entire process may then be repeated any number of times, obtaining a better approximation each time if the conditions for convergence are satisfied.

The convergence of the sequence of approximations so obtained has not been investigated rigorously, but undoubtedly could be along the lines disclosed by Schmidt in Phil. Mag., 32, 369 (1941). However, R. C. Briant of the Mellon Institute, Pittsburgh, Pa., has demonstrated the simple sufficient condition Gii lln= i, '1`,v.=1, 2 n

Any system may be reduced to this canonical form by rearrangement of the sequence of the variables and equations, and by a few simple additions or subtractions of the equations.

The mechanism of operation of the machine of 4the instant invention can be made similar to the latter method. The value of the qth variable is adjusted to satisfy the qth equation. The difference, however, is that in Jacobis method the adjustment is made on one variable at a time in turn, while the present device performs all adjustments simultaneously.

If the simultaneous adjustment of the variables does not lead to the correct solution, a slight alteration of the machine will permit it to function exactly as in Jacobis method. All that is necessary is to provide the amplifier input lines with switches 3l as shown in Figures 1 and 6, and a large condenser Ell to ground as shown in Figure 6. Starting with all the switches open; they would then be closed momentarily one at a time in turn; a motor-driven commutator, not shown, could be used to do the switching automatically. At each closing of a switch one equation would be satisfied by adjustment of the variable in its diagonal term, and a. new and better approximation would be obtained for each complete cycle of switching. Since the response of the ampliers is nearly instantaneous, very rapid switching could be used and a high order approximation obtained in a fraction of a second. The machine would, thus, still give continuous indication. The commutator can be replaced with an electronic switching device, such as a special type of multivibrator circuit, so that the instrument would have no moving parts.

Although a three-element machine (Figure l) has been briefly described in this application to illustrate, in part, the principles and operation of one form of calculating machine of which the instant invention is an improvement and a twoelement machine (Figure 9) is shown to illustrate in more detail a form of the present invention, it is to be understood that this matter is not to be construed as limiting the invention to a, machine for the solution of a system of three or less simultaneous equations, but the invention contemplates electronic calculating machines 13 adapted for use in solving a system of simultaneous algebraic equations for any number of variables up to n and machines may be built for any value of n.

The principle of employing fully degenerative feed-back for balancing electronic calculating machines in general as described above and in aforementioned application Serial No. 479,790 has broad application and it is to be understood that the applicant is not to be limited by the specific application of this principle as described above. As an additional example of the application of fully degenerative feed-back for balancing electronic calculating machines, it can be employed in machines for drawing integral solution curves, starting from any boundary point, of the general order of differential equations of the first order. Numerous other examples of Ithe use of fully degenerative feed-back might be postulated. Broadly speaking, it provides the only convenient and general means for balancing an equation electrically; it is the electrical equivalent of the mathematicians equal sign. Since all of the usual elementary operations of mathematics can Abe performed electrically, it is possible to build a machine for solving any system of equations by using such negative feed-back.

What We claim as our invention is:

1. An electronic computing device for determining the roots of a system of simultaneous linear algebraic equations with n variablesv where 'n is any positive integer greaterthan unity comprising in combination n electronic amplifiers capable of supplying a large A.C. for a small D.C. input, n D.C. voltage sources each adjustable to produce a voltage proportional to the respective constant terms of the system of equations, and n2 rectifiers each respectively supplying D. C. to the entirety of one of 'n2 potentiometers whose adjustable portions are respectively made proportional to the coefficients of the system of equations, an electrical network involving the aforesaid elements in such a way so that one of said adjusted voltage sources and n adjusted potentiometer portions all corresponding to one equation and one amplier input are in series and so that the rectiiers supplying current to each of said n series-connected potentiometer portions are respectively fed from the output of a diiferent one of said amplifiers, and currentresponsive devices in at least one of said rectier circuits for indicating the desired roots of the equations.

2. An electrical computing device for determining the roots of a system of simultaneous linear algebraic equations with n variables where n is any positive integer greater than unity comprising in combination a source of alternating current, n amplifiers connected to said A.-C. source and adapted to regulate A. C. obtained from said source in response to a small D.C. amplifier input, n D.C. voltage sources each adjustable to produce a voltage proportional to the respective constant terms of the system of equations, 112 rectiers respectively connected to and supplying D. C. to one of n2 adjustable potentiometers whose adjustable portions are respectively made proportional to the coeicients of the system of equations, an electrical network involving the aforesaid elements in such a way so that one of said adjusted voltage sources and n of said adjusted potentiometer portions and one amplifier input are in series and so that the rectiers supplying current to said n series-connected potentiometer portions are respectively fed A. C. from one of said amplifiers, and current-responsive means in at least one of said potentiometer circuits for indicating the desired roots of the equations.

3. An electronic computing device for determining the roots of a system of simultaneous linear algebraic equations with n variables where n is an integer greater than unity comprising in combination an electronic oscillator, n electronic amplifiers connected to said oscillator and adapted to regulate A. C. obtained from the oscillator in response to a small D.C. amplifier input, n D.C. voltage sources each adjustable to produce a voltage proportional to the respective constant terms of the system of equations, n2 pairs of full-wave rectifiers whose outputs are connected in series one of each pair being supplied with A. C. from the aforesaid oscillator and the other of each pair being supplied with regulated A. C. from one of the aforesaid amplifiers and the D. C. of the series-connected outputs being applied respectively to the entirety of one of 11.2 potentiometers Whose adjustable portions are respectively made proportional to the coeic'ients of the system of equations, an electrical network involving the aforesaid elements in such a Way so that one of said adjusted voltage sources and n. adjusted potentiometer portions al1 corresponding to one equation and one amplifier input are in series and so that the rectiers supplying current to said n series-connected potentiometers are respectively fed from the output of a different one of said amplifiers, and current-responsive devices in at least one of said rectifier circuits `for indicating the desired roots of the equations.

JOHN R. BOWMAN. RALPH T. STEINBACK.

REFERENCES CITED The following references are of record in the le of this patent:

UNITED STATES PATENTS Number Name Date 1,893,009 Ward Jan. 3, 1933 2,417,098 Wilcox Mar. 11, 1947 2,454,549 Brown Nov. 23, 1948 2,455,974 Brown Dec. 14, 1948 2,458,697 Fenske Jan. 11, 1949 2,459,106 Hardy Jan. 11, 1949 

